Lyapunov Stability Analysis of Higher-Order 2-D Systems

Kojima, Chiaki; Rapisarda, Paolo; Takaba, Kiyotsugu

doi:10.1007/978-3-540-93918-4_18

Cited by 2 publications

(2 citation statements)

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“…Analogous conditions for linear Roesser model one can find in El-Agizi and Fahmy (1979), Bliman (2002). In Kojima et al (2011) it is shown that asymptotic stability of linear 2-D system is equivalent to the existence of a vector Lyapunov functional satisfying certain positivity conditions together with its divergence along the system trajectories. In Tatsuhi (2001) it is, however, shown that application of 2-D Lyapunov matrix inequality is limited in application to robust stability of a system described by the Fornasini-Marchesini model.…”

Section: Introductionmentioning

confidence: 95%

Stability of nonlinear time-varying digital 2-D Fornasini-Marchesini system

Kurek

2012

Multidim Syst Sign Process

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Section: Introductionmentioning

confidence: 95%

Stability of nonlinear time-varying digital 2-D Fornasini-Marchesini system

Kurek

2012

Multidim Syst Sign Process

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“…Background material. We give only the minimum amount of information needed; see [10,17,18] for more information, and [11,13,14,21,23] for important details and for applications of 2D bilinear-and quadratic differential forms.…”

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confidence: 99%

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

Rapisarda¹,

Antoulas²

2016

SIAM J. Control Optim.

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Abstract. We solve two problems in modelling polynomial vector-exponential trajectories dependent on two independent variables. In the first one we assume that the data-generating system has no inputs, and we compute a state representation of the Most Powerful Unfalsified Model for this data. In the second instance we assume that the data-generating system is controllable and quarterplane causal, and we compute a Roesser i/s/o model. We provide procedures for solving these identification problems, both based on the factorization of constant matrices directly constructed from the data, from which state trajectories can be computed.Key words. multidimensional systems, most powerful unfalsified model, Roesser models, bilinear differential forms AMS subject classifications. 93A30, 93B15, 93B20, 93B30, 93C201. Introduction. We consider two problems in modelling two-dimensional (2D in the following) continuous trajectories from data. In both cases the data consistis of polynomial vector-exponential trajectories, and we seek state-space models explaining it, i.e. systems of partial differential equations of first order in an auxiliary, "state" variable, and zeroth-order in the measured, "external" variable. The two situations differ in the model class we assume the data-generating system belongs to: in the first case we seek an autonomous state model , i.e. a system without inputs; in the second one we assume that an input/output partition of the external variable is given, and we compute an input-state-output (i/s/o in the following) model.Modelling 2D polynomial vector-exponential trajectories with autonomous systems has been considered in [32,33], on whose results the first part of this paper on the computation of autonomous models heavily relies. Modelling vector-exponential trajectories with transfer-function (i.e. input-output) models is closely related to twovariable rational interpolation; the latter has been investigated in the SISO case in [2]. The approach taken in the present paper differs fundamentally from those: we use data to first compute state trajectories corresponding to it, and in a second stage we compute a state representation for the data and the identified state trajectories by solving linear equations in the unknown state-, input-and output matrices.Modelling methodologies where state-trajectories are computed from data and state-equations are subsequently computed are well-known in the 1D case as subspace identification methods (see e.g. [12]). Such ideas have been pursued much less frequently in the 2D case: see [7,19] for a pioneering subspace-identification approach to the computation of i/s/o representations of denominator-separable 2D discretesystems from data. Our modelling approach differs essentially from those, in that it does not exploit the shift-invariance properties of data trajectories, but rather the fact that external properties-i.e. properties at the level of external variables, in our case duality-are reflected into internal properties-i.e. at the level of state. To make the...

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Lyapunov Stability Analysis of Higher-Order 2-D Systems

Cited by 2 publications

References 19 publications

Stability of nonlinear time-varying digital 2-D Fornasini-Marchesini system

Stability of nonlinear time-varying digital 2-D Fornasini-Marchesini system

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

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